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;; GCD.
(defn gcd [a b]
(if (zero? b)
(recur b (mod a b))))
;; Square Root.
(defn sqrt [n] (Math/sqrt n))
;; Random Number Generator represented as a computational object
;; with local, mutable and independent state (that is why we used
;; atoms here).
(defn random []
(let [x (atom (rand-int 1000000000))]
(fn [] (reset! x (rand-int 1000000000)))))
;; Stream (infinite sequence) of random numbers.
(def random-numbers
(repeatedly #((random))))
;; Monte Carlo method.
(defn monte-carlo [experiment-as-stream passed failed]
(letfn [(next [passed failed]
(cons (/ passed (+ passed failed))
(lazy-seq (monte-carlo (rest experiment-as-stream)
(if (first experiment-as-stream)
(next (inc passed) failed)
(next passed (inc failed)))))
;; Experiment - approximating `pi` with Caesaro test.
;; Stream (infinite sequence) of coprime numbers.
(defn map-successive-pairs [f stream]
(cons (f (first stream) (first (rest stream)))
(lazy-seq (map-successive-pairs f (rest (rest stream))))))
(def caesaro-stream
(map-successive-pairs #(= (gcd %1 %2) 1) random-numbers))
(def pi
(map #(sqrt (/ 6.0 %))
(monte-carlo caesaro-stream 0 0)))
(println (last (take 1000000 pi)))